Learner Guide

Cambridge International AS & A Level

Mathematics 9709
For examination from 2020

Version 1
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Contents
C

Section 1: About this guide ............................................................................................................................ 4

Section 2: Syllabus content – what you need to know ................................................................................ 5
Prior knowledge ............................................................................................................................................. 6
Aims ............................................................................................................... Error! Bookmark not defined.
Key concepts ................................................................................................................................................. 6
Section 3: How you will be assessed ............................................................................................................ 7
About the examinations ................................................................................................................................. 7
About the papers ........................................................................................................................................... 8
Information about each paper ........................................................................ Error! Bookmark not defined.
Section 4: What skills will be assessed ...................................................................................................... 10
Section 5: Command words ......................................................................................................................... 11
Section 6: Example candidate response ..................................................................................................... 12
A. Question............................................................................................................................................... 13
B. Mark scheme ....................................................................................................................................... 14
C. Example candidate response and examiner comments ...................................................................... 15
D. Common mistakes .............. . ... ..16

E. General advice .. ... . .16

Section 7: Revision........................................................................................................................................ 17
General advice ............................................................................................................................................. 17
Paper 1 advice ............................................................................................................................................. 20
Paper 2 advice ............................................................................................................................................. 20
Paper 3 advice ............................................................................................................................................. 20
Paper 4 advice ............................................................................................................................................. 21
Paper 5 advice ............................................................................................................................................. 21
Paper 6 advice ............................................................................................................................................. 22
Revision checklists ...................................................................................................................................... 22
Section 8: Useful websites ........................................................................................................................... 52
Learner Guide

Section 1: About this guide

This guide explains what you need to know about your Cambridge International AS & A Level Mathematics
course and examinations.

This guide will help you to:

• understand what skills you should develop by taking this AS & A Level course

• understand how you will be assessed

• understand what we are looking for in the answers you write

• plan your revision programme

• revise, by providing revision tips and an interactive revision checklist (Section 7).

Following a Cambridge International AS & A level programme will help you to develop abilities that
universities value highly, including a deep understanding of your subject; higher order thinking skills
(analysis, critical thinking, problem solving); presenting ordered and coherent arguments; and independent
learning and research.

Studying Cambridge International AS & A Level Mathematics will help you to develop a set of transferable
skills, including the ability to work with mathematical information; think logically and independently; consider
accuracy; model situations mathematically; analyse results and reflect on findings.

Our approach to this course should encourage you to be:

confident, using and sharing information and ideas, and using

mathematical techniques to solve problems. These skills build
confidence and support work in other subject areas as well as in
mathematics.

responsible, through learning and applying skills that prepare you

for future academic studies, helping you to become numerate
members of society.

reflective, through making connections between different branches

of mathematics and considering the outcomes of mathematical
problems and modelling.

innovative, through solving both familiar and unfamiliar problems in

different ways, selecting from a range of mathematical and problem
solving techniques.

engaged, by the beauty and structure of mathematics, its patterns

and its many applications to real life situations.

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Section 2: Syllabus content – what you need to know

This section gives you an outline of the syllabus content for this course. There are six components that
can be combined in specific ways (see Section 3). Talk to your teacher to make sure you know which
components you will be taking.

Content section Assessment Topics included

component

1 Pure Mathematics 1 Paper 1 1.1 Quadratics

1.2 Functions
1.3 Coordinate geometry
1.4 Circular measure
1.5 Trigonometry
1.6 Series
1.7 Differentiation
1.8 Integration

2 Pure Mathematics 2 Paper 2 2.1 Algebra

(AS only) 2.2 Logarithmic and exponential functions
2.3 Trigonometry
2.4 Differentiation
2.5 Integration
2.6 Numerical solution of equations

3 Pure Mathematics 3 Paper 3 3.1 Algebra

3.2 Logarithmic and exponential functions
3.3 Trigonometry
3.4 Differentiation
3.5 Integration
3.6 Numerical solution of equations
3.7 Vectors
3.8 Differential equations
3.9 Complex numbers

4 Mechanics Paper 4 4.1 Forces and equilibrium

4.2 Kinematics of motion in a straight line
4.3 Momentum
4.4 Newton’s laws of motion
4.5 Energy, work and power

5 Probability & Statistics 1 Paper 5 5.1 Representation of data

5.2 Permutations and combinations
5.3 Probability
5.4 Discrete random variables
5.5 The normal distribution

6 Probability & Statistics 2 Paper 6 6.1 The Poisson distribution

6.2 Linear combinations of random variables
6.3 Continuous random variables
6.4 Sampling and estimation
6.5 Hypothesis tests

Make sure you always check the latest syllabus, which is available from our public website. This will also
explain the different combinations of components you can take.
Cambridge International AS & A Level Mathematics 9709 5
Learner Guide

Prior knowledge
Knowledge of the content of the Cambridge IGCSETM Mathematics 0580 (Extended curriculum), or
Cambridge International O Level (4024/4029), is assumed.

Key concepts
Key concepts are essential ideas that help you to develop a deep understanding of your subject and
make links between different aspects of the course. The key concepts for Cambridge International AS & A
Level Mathematics are:

• Problem solving
Mathematics is fundamentally problem solving and representing systems and models in different
ways. These include:

• Algebra: this is an essential tool which supports and expresses mathematical reasoning
and provides a means to generalise across a number of contexts.

• Geometrical techniques: algebraic representations also describe a spatial relationship,

which gives us a new way to understand a situation.

• Calculus: this is a fundamental element which describes change in dynamic situations

and underlines the links between functions and graphs.

• Mechanical models: these explain and predict how particles and objects move or remain
stable under the influence of forces.

• Statistical models: these are used to quantify and model aspects of the world around us.
Probability theory predicts how chance events might proceed, and whether assumptions
about chance are justified by evidence.

• Communication
Mathematical proof and reasoning is expressed using algebra and notation so that others can
follow each line of reasoning and confirm its completeness and accuracy. Mathematical notation
is universal. Each solution is structured, but proof and problem solving also invite creative and
original thinking.

• Mathematical modelling
Mathematical modelling can be applied to many different situations and problems, leading to
predictions and solutions. A variety of mathematical content areas and techniques may be
required to create the model. Once the model has been created and applied, the results can be
interpreted to give predictions and information about the real world.

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Section 3: How you will be assessed

Cambridge International AS Mathematics makes up the first half of the Cambridge International A Level
course in mathematics and provides a foundation for the study of mathematics at Cambridge International A
Level.

About the examinations

There are three different combinations of papers you can take to obtain an AS level Mathematics
qualification:

• Papers 1 and 2 (this cannot lead to an A Level route)

• Papers 1 and 4
• Papers 1 and 5

There are two different combinations of papers you can take to obtain an A Level Mathematics qualification:

• Papers 1, 3, 4 and 5
• Papers 1, 3, 5 and 6.

These are summarised in the diagram. Find out from your teacher which papers you will be taking.

Your knowledge will build as you progress through the course. Paper 1 Pure Mathematics 1 is the foundation
for all other components. (Solid lines mean there is direct dependency of one paper on another; dashed lines
mean that prior knowledge from previous paper is assumed.)

Paper 2

Paper 1
Paper 3

Paper 4

Paper 5 Paper 6

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About the papers

The table gives you further information about the examination papers:

Time and
Component Questions Percentage of total mark
marks

Paper 1 1 hour 50 A written paper with 10 to 12 60% of AS Level

minutes structured questions based on the
Pure subject content for Pure Mathematics 30% of A Level
Mathematics 1 (75 marks) 1. You must answer all questions.

Paper 2 1 hour 15 A written paper with 6 to 8 structured 40% of AS level

minutes questions based on the subject
Pure content for Pure Mathematics 2. You (not offered as part of A Level)
Mathematics 2 (50 marks) must answer all questions.

Paper 3 1 hour 50 A written paper with 9 to 11 (not offered as part of AS Level;

minutes structured questions based on the compulsory for A Level)
Pure subject content for Pure Mathematics
Mathematics 3 (75 marks) 3. You must answer all questions. 30% of A Level

Paper 4 1 hour 15 A written paper with 6 to 8 structured 40% of AS level

minutes questions based on the subject
Mechanics content for Mechanics. You must 20% of A Level
(50 marks) answer all questions.

Paper 5 1 hour 15 A written paper with 6 to 8 structured 40% of AS level

minutes questions based on the subject
Probability & content for Probability & Statistics 1. 20% of A Level
Statistics 1 (50 marks) You must answer all questions.

Paper 6 1 hour 15 A written paper with 6 to 8 structured (not offered as part of AS Level)
minutes questions based on the subject
Probability & content for Probability & Statistics 2. 20% of A Level
Statistics 2 (50 marks) You must answer all questions.

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 Learner Guide

Below is a typical page taken from one of the papers.

The number of marks

available for each part
of a question is
indicated.

A structured You answer on the question paper.

question means that
the question is split
into parts. The parts
are labelled (a), (b),
(c), etc. and these
may have sub-parts
(i), (ii), (iii), etc. All working should be shown neatly
Questions are of and clearly in the spaces provided for
varied lengths. each question.

New questions often start on a fresh

page, so more answer space may be
provided than is needed.

If you need more space, you should

use the lined page at the end of the
question paper. Make sure you
clearly show which question(s)
you’re answering there.

The main focus of examination questions

will be the AS & A Level Mathematics
subject content. However, in examination
questions, you might need to make use of
prior knowledge and mathematical
techniques from previous study.

Some questions might require you to

sketch graphs or diagrams, or draw
accurate graphs. You should use a calculator where
appropriate.

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Learner Guide

Section 4: What skills will be assessed

The examiners take account of the following skills areas (assessment objectives) in the examinations:

AO1 Knowledge and understanding AO2 Application and communication

Assessment objectives (AO) What does the AO mean?

AO1 Knowledge & understanding Demonstrating that you understand what is required of a question.
• Show understanding of relevant Remembering methods and notation.
mathematical concepts,
terminology and notation Showing your working and the methods you used will help you
demonstrate your AO1 skills.
• Recall accurately and use
appropriate mathematical
manipulative techniques

AO2 Application & communication Deciding which methods to use when solving a problem, and
• Recognise the appropriate how/when to apply more than one method. You need to write your
mathematical procedure for a solution clearly and logically so that someone else can understand it.
given situation
• Apply appropriate combinations Showing your working and the methods you used will help you
of mathematical skills and demonstrate your AO2 skills.
techniques in solving problems
• Present relevant mathematical
work, and communicate
corresponding conclusions, in a
clear and logical way

It is important that you know the different weightings (%) of the assessment objectives, as this affects how
the examiner will assess your work.

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Section 5: Command words

The table below includes command words used in the assessment for this syllabus. The use of the
command word will relate to the subject context.

You should also be aware of the term ‘Hence’. If a question part includes the key word ‘hence’, it means that
you must use your work from the previous part to help you answer the question part you are now on.

‘Hence or otherwise’ means that you can either use your work from the previous part to help answer the
question part you are now on (this is usually the easiest option) or you can use another method of your
choice.

So, ‘Hence’ in a mathematical context is often there to help you with the next step in a question, but is also
there if a particular method of solution is required.

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Learner Guide

Section 6: Example candidate response

This section takes you through an example question and candidate response. It will help you to see how to
identify the command words within questions and to understand what is required in your response.
Understanding the questions will help you to know what you need to do with your knowledge. For example,
you might need to state something, calculate something, find something or show something.

All information and advice in this section is specific to the example question and response being
demonstrated. It should give you an idea of how your responses might be viewed by an examiner but
it is not a list of what to do in all questions. In your own examination, you will need to pay careful
attention to what each question is asking you to do.

This section is structured as follows.

Question
Command words in the question have been
highlighted and their meaning explained.
This should help you to understand clearly
what is required by the question.

Mark scheme
This tells you as clearly as possible what an
examiner expects from an answer in order
to award marks.

Example candidate response

This is a sample answer of a high standard.
Points have been highlighted to show you
how to answer a question.

General advice
These tips will help you to answer
questions in general.

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Question
The question used in this example is from Paper 1 Pure Mathematics 1. The answer spaces have been
removed from the layout below so that it all fits on one page.

12

The diagram shows the curve with equation y = x(x – 2)2. The minimum point on the curve has
coordinates (a, 0) and the x-coordinate of the maximum point is b, where a and b are constants.

State: The examiner will be expecting

you to write down a value of without
any working.

(a) State the value of a. [1]

Calculate: The examiner will be expecting you to
apply differentiation methods to work out (the -
coordinate of the maximum) in part (b) and (the
(b) Calculate the value of b. [4]
minimum gradient of the curve) in part (d).

(c) Find the area of the shaded region. [4]

dy
(d) The gradient, , of the curve has a minimum value m. Calculate the value of m. [4]
dx

Find is a key instruction word. The

examiner will be expecting you to
apply integration methods to find the
area of the shaded region in part (c).

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Learner Guide

Mark scheme
For each Paper, there is a detailed mark scheme showing how the various types of mark are allocated by the
examiners for each question: M (method mark), A (accuracy mark), B (mark for a correct statement or step),
DM (dependent method mark) and DB (dependent B mark). The mark scheme shows the final answers for
each part of a question and, when appropriate, the lines of working you need to show to reach that answer.

Method marks: When method marks are available, you can get these for certain lines in the working that
leads to the final answer. This means that, even if you get the final answer incorrect, you can still get some
marks for your working. The mark scheme will often give more than one method but it will not include all
possible methods. If your method is not in the mark scheme, but it is accurate and relevant, the examiner will
award you marks for the appropriate parts of your working.

Final answer: This value is what the examiner expects to see. For some questions, the answer has to be
exactly as given in the mark scheme. For other questions there will be acceptable alternatives to the given
value. In both cases, it will be clearly stated in the mark scheme. Alternatives to the final answer might be
allowed, for example, where the question involves continuous data or when measuring or rounding values.

Below is the mark scheme for Question12.

Partial
Question Answer Guidance
Marks
12(a) =2 B1
B marks are ‘independent marks’.
Example of a final answer. This B1 is awarded if the value for
is seen, even without any working.

Total mark available for part (a) [1]

12(b) = 4 +4 B1

=3 8 +4 B2FT FT B1 for 3
B1 for 8 + 4

FT = follow through marks. Here, if

you expanded the function
incorrectly, you can still score the
B1FT mark for correctly
differentiating your cubic function.

2 B1 Dependent on method seen for solving

( 2)(3 2) = 0 =
3 quadratic equation.
You cannot get the mark without
showing the working, as you need to
demonstrate that you know how to
solve a quadratic equation; you can’t
than just use a calculator.

Total mark available for part (b) [4]

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 Learner Guide

Partial
Question Answer Guidance
Marks
12(c) 4 B2
Area = d = +2 B1 for
4 3
B1 for +2

One B1 mark is awarded for

each correct expression.

4 +8 FT M1FT their a from (a)

The method mark is awarded for

substituting in the limits. Here, this
means that you can get this
method mark even if your upper
limit is incorrect, so long as you
have correctly used your value for
from part (a).

4 A1 Unsupported answer receives 0 marks

3
A correct answer with no method
shown, or resulting from an
incorrect method, gets 0 marks.

Total mark available for part (c) [4]

12(d) =6 8=0 = M1*A1 Attempt 2nd derivative and set = 0

M1 is awarded for attempting to

find the second derivative and
Example of
working that is forming the equation = 0. A1 is
awarded a awarded if the working is all correct
method mark. leading to correct -value.

4 DM1A1
When = , DM1 is a method mark that is
3
d 4 dependent on having scored the
(or )=
d 3 M1 mark indicated by the * in part
(d). A1 is awarded for getting the
correct minimum value of .

Total mark available for part (d) [4]

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Learner Guide

Now let’s look at the example candidate response to the question and the examiner comments.

Example candidate response

This model answer is awarded full marks, i.e. 13 out of 13. The Examiner comments are in the boxes.

‘State’ is a clue that not much work is needed and you

can just write down the answer. You know that y = 0 at
this point and the function is already factorised.

Expand expression ( 2) to
obtain the correct cubic

Differentiate all of the terms.

Set = 0 and solve an equation

to find b, the x-coordinate of the
maximum. You need to show how
you solved the equation to find the
two values, then write down b. It is
not enough to solve the quadratic
equation on your calculator and
write down the answers, although
you can use your calculator to
check once you have done the
working on the Paper.

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 Learner Guide

Integrate your expression for y, using

correct notation.

Substitute in the limits, 0 and your value

of a, to get the method mark. When you
think about which limits to use, you will
Obtain the correct answer probably realise that your value for is
(you need to show the one of them. The other one is 0 and not
method for finding it). . Reading the question carefully and
thinking about the graph will help you. If it
helps, you could also label the graph with
the values you have found.

To see where is a maximum,

find and write =0

Solve the equation to find x.

Substitute this value into .

Obtain the correct value of m.

Some learners might find part (d) a little unfamiliar. Don’t be tempted to repeat what
you’ve already done or substitute values into the 2nd derivative. It is better to stop and
think about what you need to know. You are trying to analyse the gradient to see
where it is a minimum. Just as the 1st derivative tells you about a minimum in the
original function, y , so the 2nd derivative tells you about a minimum in the 1st
derivative (it’s the gradient of the gradient function).

So one way to answer this is to put = 0, solve this for and then use that value of
to find (or ). This is the method shown in the mark scheme. Another valid way of
answering this question is to complete the square on . This gives you 3 .

You can see that the minimum value ( ) would be because is always 0.
There are other valid ways of answering the question too. All of these would allow you
to gain M marks and, if the answers are correct, A marks too.

Cambridge International AS & A Level Mathematics 9709 17

Learner Guide

General advice
It is always a good idea to read the question carefully, noticing the command words and key instructions (in
this case ‘State’, ‘Find’ and ‘Calculate’). You may want to underline them to help you think what they mean.
Many candidates jump straight into writing their working only to realise too late that they’ve used the wrong
method. Read the question first and pause to think about what you need to find before you start doing any
working – this will help you to choose an efficient method so you don’t waste time in the examination. Don’t
forget that your working is part of your solution and you can gain method marks even if you don’t get as far
as a correct answer. In the example question, there are several FT (follow through) marks as well. This
means that if you can go on to use your values correctly, even if they are wrong, you can still get subsequent
method marks.

When answering a question based on a graph, such as the example question, it is often helpful to add to the
graph any values you find, or information from the question. This will help you to think about what methods
you can use to answer the question. If the question doesn’t provide a graph or diagram, it is often useful for
you to sketch your own.

Using correct notation in your working will help you to think clearly as well as making it easier for someone
else to understand what you have done.

Make sure you are clear if you need to differentiate or integrate. It is surprisingly common for learners to get
confused when a question requires both methods.

If you have had a good attempt at a question and still not managed to finish it, it is best to move on to
another question and come back to it later. This will help you to make good use of the time you have
available.

Allow a few minutes at the end of the examination to check your work. This will help you to spot errors in
your arithmetic that could lose you marks.

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Section 7: Revision

This advice will help you revise and prepare for the examinations. It is divided into general advice and
specific advice for each of the papers.

Use the tick boxes to keep a record of what you have done, what you plan to do or what you understand.

General advice
Before the examination

Find out when the examinations are and plan your revision so you have enough time for each
topic. A revision timetable will help you.

Find out how long each Paper is and how many questions you have to answer.

Know the meaning of the command words used in questions and how to apply them to the
information given. Highlight the command words in past papers and check what they mean.
There is a list on p11 of this guide.

Make revision notes; try different styles of notes. See the Learner Guide: Planning, Reflection
and Revision (www.cambridgeinternational.org/images/371937-learner-guide-planning-
reflection-and-revision.pdf) which has ideas about note-taking. Discover what works best for
you.

Work for short periods then have a break. Revise small sections of the syllabus at a time.

Build your confidence by practising questions on each of the topics.

Make sure you practise lots of past examination questions so that you are familiar with the
format of the examination papers. You could time yourself when doing a paper so that you
know how quickly you need to work in the real examination.

Look at mark schemes to help you to understand how the marks are awarded for each
question.

Make sure you are familiar with the mathematical notation that you need for this syllabus. Your
teacher will be able to advise you on what is expected.

Check which formulae are in the formula booklet available in the examination, and which ones
you need to learn.

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Learner Guide

During the examination

Read the instructions carefully and answer all the questions.

Check the number of marks for each question or part question. This helps you to judge how
long you should be spend on the response. You don’t want to spend too long on some
questions and then run out of time at the end.

Do not leave out questions or parts of questions. Remember, no answer means no mark.

You do not have to answer the questions in the order they are printed in the answer booklet.
You may be able to do a later question more easily then come back to an earlier one for
another try.

Read each question very carefully. Misreading a question can cost you marks:
o Identify the command words – you could underline or highlight them.
o Identify the other key words and perhaps underline them too.
o Try to put the question into your own words to understand what it is really
asking.

Read all parts of a question before starting your answer. Think carefully about what is
needed for each part. You will not need to repeat material.

Look very carefully at the information you are given.

o For graphs, read the title, key, axes, etc. to find out exactly what they show.
o For diagrams, look at any angles and lengths.
o Try using coloured pencils or pens to pick out anything that the question
asks you about.

Answer the question. This is very important!

o Use your knowledge and understanding.
o Do not just try all the methods you know. Only use the ones you need to
answer the question.

Make sure that you have answered everything that a question asks. Sometimes one
sentence asks two things, e.g. ‘Show that and hence find ’. It is easy to concentrate on
the first request and forget about the second one.

Always show your working. Marks are usually awarded for using correct steps in the method
even if you make a mistake somewhere.

Don’t cross out any working until you have replaced it by trying again. Even if you know it’s
not correct you may still be able to get method marks. If you have made two or more
attempts, make sure you cross out all except the one you want marked.

Use mathematical terms in your answers when possible.

Annotated diagrams and graphs can help you, and can be used to support your answer. Use
them whenever possible but do not repeat the information in words.

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During the examination

Make sure all your numbers are clear, for example make sure your ‘1’ doesn’t look like a ‘7’.

If you need to change a word or a number, or even a sign (+ to – for example), it is better to
cross out your work and rewrite it. Don’t try to write over the top of your previous work as it
will be difficult to read and you may not get the marks.

Don’t write your answers in two columns in the examination. It is difficult for the examiners to
read and follow your working.

Advice for all Papers

Give numerical answers correct to 3 significant figures (3SF) in questions where no accuracy
is specified, except for angles in degrees, which need to be to 1 decimal place (1 d.p.).

Make sure you know the difference between 3 significant figures and 3 decimal places, e.g.
0.03456 is 0.0346 (3 SF) but is 0.035 (3 d.p.). You would not get the mark for 0.035 as it is not
accurate enough.

For your answers to be accurate to 3SF, you will have to work with at least 4SF throughout the
question. For a calculation with several stages, it is usually best to use all the figures in your
calculator. However, you do not need to write all these figures down in your working but you
should write down each to at least 4SF before rounding your final answer to 3SF.

If a question specifies how accurate your answer needs to be, you must give your final answer
to that degree of accuracy. In questions where the accuracy is not specified, however, you will
not be penalised if you give answers that are more accurate than 3SF.

Some questions ask for answers in exact form. In these questions you must not use your
calculator to evaluate answers and you must show the steps in your working. Exact answers
may include fractions or square roots and you should simplify them as far as possible.

You are expected to use a scientific calculator in all your examination papers. You are not
allowed to use computers, graphical calculators and calculators capable of symbolic algebraic
manipulation or symbolic differentiation or integration.

Check that your calculator is in degree mode for questions that have angles in degrees and in
radian mode for questions that have angles in radians.

In questions where you have to show an answer that is given, it is particularly important to show
all your working. To gain the marks, you need to convince the examiner that you understand all
the steps in getting to the answer.

Often the ‘given’ answer from a ‘show that’ question is needed in the next part of the question.
This means that even if you couldn’t show how to obtain it, you can still carry on with other parts
of the question. In later parts that rely on the ‘given’ answer, you should always use the ‘given’
answer exactly as it is stated in the question, even if you have obtained a different answer
yourself.

There are no marks available for just stating a method or a formula. You have to apply the
method to the particular question, or use the formula by substituting values in.

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Advice for Paper 1, 2 and 3

Make sure you know all the formulae that you need (even ones from Cambridge ICGSE or O
Level). If you use an incorrect formula you will score no marks.

Check to see if your answer is required in exact form. In a trigonometry question you will need to
use exact values of sin 60°, for example, to obtain an exact answer. Make sure you know the
exact values of sin, cos and tan of 30°, 45° and 60° as they are not provided in the examination.

Calculus questions (Paper 2 and 3) involving trigonometric functions use values in radians, so
make sure your calculator is in the correct mode if you need to use it.

Paper 4 advice
When you need a numerical value for ‘g’, use 10 ms-2 (unless the question states otherwise).

Always draw clear force diagrams, whether the question asks for them or not. This will help you
to solve problems.

Make sure you are familiar with common words such as ‘initial’, ‘resultant’, ‘smooth’, ‘rough’,
‘light’ and ‘inextensible’. Also make sure that you know the difference between ‘mass’ and
‘weight’.

Go through some past examination questions and highlight common words and phrases. Learn
what they mean and make sure you can recognise them.

Make sure you know the algebraic methods from Paper 1 Pure Mathematics 1.

Make sure you know these trigonometrical results:

sin
sin( 90° - °) = cos °, cos( 90° - °) = sin °, tan , sin2 + cos2 = 1 .
cos

When a question mentions bodies in a ‘realistic’ context, you should still treat them as particles.

Vector notation will not be used.

Advice for Paper 5 and 6

You can give probabilities as fractions, decimals or percentages in your answer.

Always look to see if your answer makes sense. For example, if you calculated a probability as
1.2, you would know you had made a mistake and should check your solution.

Sometimes you might be asked to give an answer ‘in the context of the question’. This means
that your answer must use information about the situation described in the question. For
example, don’t just write ‘The events must be independent’ (which could apply in any situation).
If the question is about scores on a dice for instance, you could write ‘The scores when the dice
is rolled must be independent’. If the question is about the time taken by people to carry out a
task, you could write ‘The times taken by the people must be independent of each other’.

When you are answering a question about a normal distribution, it is useful to draw a diagram;
this can help you to spot errors. For example, if you are finding a probability you will be able to
see from a diagram if you would expect the answer to be greater or less than 0.5.

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Paper 6 advice
When you are carrying out a hypothesis test, you should always state the conclusion in the
context of the question. Don’t state conclusions in a way that implies that a hypothesis test has
proved something. For example, it is better to write ‘There is evidence that the mean weight of
the fruit has increased’ instead of ‘The test shows that the mean weight of the fruit has
increased’.

When you are calculating a confidence interval, it may not always be sensible to apply the usual
3 SF rule about accuracy. For example, if an interval for a population mean works out as
(99.974, 100.316), it doesn’t make sense to round both figures to 100. In this case, it is better to
give 2 d.p. or 3 d.p so that the width of the interval is more accurate.

Revision checklists
The tables below can be used as a revision checklist: It doesn’t contain all the detailed knowledge you
need to know, just an overview. For more detail see the syllabus and talk to your teacher.

You can use the tick boxes in the RAG checklist to show when you have revised and are happy that you do
not need to return to it. Tick the ‘R’, ‘A’, and ‘G’ column to record your progress. The ‘R’, ‘A’ and ‘G’ represent
different levels of confidence, as follows:

• R = RED means you are really unsure and lack confidence in that area; you might want to focus your
revision here and possibly talk to your teacher for help
• A = AMBER means you are reasonably confident in a topic but need some extra practice
• G = GREEN means you are very confident in a topic

As your revision progresses, you can concentrate on the RED and AMBER topics, in order to turn them into
GREEN topics. You might find it helpful to highlight each topic in red, orange or green to help you prioritise.

You can use the ‘Comments’ column to:

• add more information about the details for each point
• add formulae or notes
• include a reference to a useful resource
• highlight areas of difficulty or things that you need to talk to your teacher about or look up in a textbook.

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Paper 1 Pure Mathematics 1

Topic You should be able to R A G Comments

1.1 Quadratics carry out the process of completing the square for a quadratic
polynomial ax2 + bx + c and use a completed square form

e.g. to locate the vertex of the graph y - ax2 _ bx + c, or to

sketch the graph

find the discriminant of a quadratic polynomial ax 2 + bx + c

and use the discriminant

e.g. to determine the number of real roots of the equation

ax 2 + bx + c = 0 . Knowledge of the term ‘repeated root’ is
included

solve quadratic equations, and quadratic inequalities, in one

unknown

By factorising, completing the square and using the formula

solve by substitution a pair of simultaneous equations of

which one is linear and one is quadratic

e.g. x + y + 1 = 0, x 2 + y 2 = 25 , 2 x + 3 y = 7, 3 x 2 = 4 + 4 xy

recognise and solve equations in x which are quadratic in

some function of x.

e.g. x 4 − 5 x 2 + 4 = 0 , 6 x + x − 1 = 0 , tan2 x = 1 + tan x

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1.2 Functions understand the terms function, domain, range, one-one

function, inverse function and composition of functions

identify the range of a given function in simple cases, and

find the composition of two given functions

1
Including e.g. range of f : x a for x ≥ 1 and range of
x
g : x a x 2 + 1 for x ∈ R . Including the condition that a
composite function gf can only be formed when the range of f
is within the domain of g

determine whether or not a given function is one-one, and

find the inverse of a one-one function in simple cases

Including e.g. finding the inverse of

h : x a (2 x + 3)2 − 4 for x < − 32

illustrate in graphical terms the relation between a one-one

function and its inverse

Indication of the mirror line y = x will be expected in

sketches

understand and use the transformations of the graph of y =

f(x) given by y = f(x) + a, y = f(x + a),
y = af(x), y = f(ax) and simple combinations of these

Use of the terms ‘translation’, ‘reflection’ and ‘stretch’ in

describing transformations is included. Questions may
involve algebraic or trigonometric functions, or other graphs
with given features

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Topic You should be able to R A G Comments

1.3 Coordinate find the equation of a straight line given sufficient information
geometry
e.g. given two points, or one point and the gradient

interpret and use any of the forms y = mx + c ,

y − y1 = m( x − x1 ) , ax + by + c = 0 in solving problems

Problems may involve calculations of distances, gradients,

mid-points, points of intersection and use of the relationship
between the gradients of parallel and perpendicular lines

understand that the equation ( x − a )2 + ( y − b )2 = r 2

represents the circle with centre (a, b) and radius r

Including use of the expanded form

x 2 + y 2 + 2gx + 2fy + c = 0

use algebraic methods to solve problems involving lines and

circles

Questions may require use of elementary geometrical

properties of circles, e.g. tangent perpendicular to radius,
angle in a semicircle, symmetry. Implicit differentiation is not
included

understand the relationship between a graph and its

associated algebraic equation, and use the relationship
between points of intersection of graphs and solutions of
equations

e.g. to determine the set of values of k for which the line

y = x + k intersects, touches or does not meet a quadratic
curve

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Topic You should be able to R A G Comments

1.4 Circular understand the definition of a radian, and use the relationship
measure between radians and degrees

use the formulae s = r θ and A = 1 r 2θ in solving problems

2
concerning the arc length and sector area of a circle

Problems may involve calculation of lengths and angles in

triangles and areas of triangles.

1.5 Sketch and use graphs of the sine, cosine and tangent
Trigonometry functions (for angles of any size, and using either degrees or
radians)
(
Including e.g. y = 3 sin x, y = 1 − cos 2 x , y = tan x + 41 π )
use the exact values of the sine, cosine and tangent of 30°,
45°, 60°, and related angles

1
e.g. cos150° = − 21 3 , sin 34 π =
2

use the notations sin−1 x , cos−1 x , tan−1 x to denote the

principal values of the inverse trigonometric relations

No specialised knowledge of these functions is required, but

understanding of them as examples of inverse functions is
expected

sinθ
use the identities ≡ tanθ and sin2 θ + cos2 θ ≡ 1
cos θ

e.g. in proving identities, simplifying expressions and solving

equations

find all the solutions of simple trigonometrical equations lying

in a specified interval (general forms of solution are not

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Topic You should be able to R A G Comments

included)

e.g. solve 3 sin 2 x + 1 = 0 for −π < x < π ,

2
3 sin θ - 5 cos θ -1 = 0 for 0 ≤ θ ≤ 360
o o

1.6 Series use the expansion of (a + b )n , where n is a positive integer

Knowledge of the greatest term and properties of the

coefficients are
not required, but the notations
n
r ()
and n! are included

recognise arithmetic and geometric progressions

use the formulae for the nth term and for the sum of the first n
terms to solve problems involving arithmetic or geometric
progressions

Problems may involve more than one progression.

Knowledge that numbers a, b, c are ‘in arithmetic
progression’ if 2b = a + c (or equivalent) and are ‘in
geometric progression’ if b 2 = ac (or equivalent) is included

use the condition for the convergence of a geometric

progression, and the formula for the sum to infinity of a
convergent geometric progression

1.7 understand the gradient of a curve at a point as the limit of

Differentiation the gradients of a suitable sequence of chords, and use the
dy d2 y
notations f ′( x ) , f ′′(x) , , and for first and second
dx dx 2
derivatives

Only an informal understanding of the idea of a limit is

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Topic You should be able to R A G Comments

expected.

e.g. includes consideration of the gradient of the chord joining

the points with x coordinates 2 and (2 + h) on the curve
3
y = x . Formal use of the general method of differentiation
from first principles is not required.

use the derivative of x n (for any rational n), together with

constant multiples, sums, differences of functions, and of
composite functions using the chain rule

dy
e.g. find given y = 2 x 3 + 5
dx

apply differentiation to gradients, tangents and normals,

increasing and decreasing functions and rates of change

Connected rates of change are included; e.g. given the rate

of increase of the radius of a circle, find the rate of increase
of the area for a specific value of one of the variables

locate stationary points and determine their nature, and use

information about stationary points in sketching graphs

Use of the second derivative for identifying maxima and

minima is included; alternatives may be used in questions
where no method is specified. Knowledge of points of
inflexion is not included

1.8 Integration understand integration as the reverse process of

differentiation, and integrate (ax + b )n (for any rational n
except −1 ), together with constant multiples, sums and
differences

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Topic You should be able to R A G Comments

⌠ 1
∫ (2x
3
e.g. − 5 x + 1)dx , dx
⌡ (2 x + 3)2

solve problems involving the evaluation of a constant of

integration

e.g. to find the equation of the curve through (1, –2) for which
dy
= 2x + 1
dx

evaluate definite integrals

1 −1
Simple cases of ‘improper’ integrals, such as ∫0 x2
dx and
∞
∫1 x −2 dx , are included

use definite integration to find

• the area of a region bounded by a curve and lines
parallel to the axes, or between a curve and a line or
between two curves,
• a volume of revolution about one of the axes

A volume of revolution may involve a region not bounded by

the axis of rotation, e.g. the region between y = 9 − x 2 and
y = 5 rotated about the x-axis

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Paper 2 Pure Mathematics 2

Topic You should be able to R A G Comments

2.1 Algebra understand the meaning of | x | , sketch the graph of y = |ax +

b| and use relations such as | a | = | b | ⇔ a 2 = b 2 and
| x − a | < b ⇔ a − b < x < a + b in the course of solving
equations and inequalities

Graphs of y = | f( x ) | and y = f(| x |) for non-linear functions f

are not included.

e.g. 3x − 2 = 2x + 7 , 2x + 5 < x + 1

divide a polynomial, of degree not exceeding 4, by a linear or

quadratic polynomial, and identify the quotient and remainder
(which may be zero)

use the factor theorem and the remainder theorem

e.g. to find factors and remainders, solve polynomial

equations or evaluate unknown coefficients. Factors of the
form (ax + b ) in which the coefficient of x is not unity are
included, and similarly for calculation of remainders

2.2 Logarithmic understand the relationship between logarithms and indices,

and exponential and use the laws of logarithms (excluding change of base)
functions
understand the definition and properties of e x and ln x ,
including their relationship as inverse functions and their
graphs

Knowledge of the graph of y = ekx for both positive and

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Topic You should be able to R A G Comments

negative values of k is included

use logarithms to solve equations and inequalities in which

the unknown appears in indices

e.g. 2 x < 5 , 3 × 23 x −1 < 5 , 3 x +1 = 42 x −1

use logarithms to transform a given relationship to linear

form, and hence determine unknown constants by
considering the gradient and/or intercept

e.g. y = kx n , y = k (a x )

2.3 understand the relationship of the secant, cosecant and

Trigonometry cotangent functions to cosine, sine and tangent, and use
properties and graphs of all six trigonometric functions for
angles of any magnitude

use trigonometrical identities for the simplification and exact

evaluation of expressions and in the course of solving
equations, and select an identity or identities appropriate to
the context, showing familiarity in particular with the use of
• sec 2 θ ≡ 1 + tan2 θ and cosec 2 θ ≡ 1 + cot 2 θ ,
• the expansions of sin( A ± B ) , cos( A ± B ) and
tan( A ± B ) ,
• the formulae for sin 2A , cos 2A and tan 2A ,
• the expression of a sinθ + b cos θ in the forms
R sin(θ ± α ) and R cos(θ ± α )

e.g. simplifying cos( x − 30°) − 3 sin( x − 60°) .

e.g. solving tanθ + cot θ = 4 or 2 sec 2 θ − tanθ = 5 or

3 cos θ + 2 sinθ = 1

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2.4 use the derivatives of e x , ln x , sin x , cos x , tan x , together

Differentiation with constant multiples, sums, differences and composites

differentiate products and quotients

2x − 4 2
e.g. , x 2 ln x , x e1− x
3x + 2

find and use the first derivative of a function which is defined

parametrically or implicitly

e.g. x = t − e2t , y = t + e2t

e.g. x 2 + y 2 = xy + 7

Including use in problems involving tangents and normals

2.5 Integration extend the idea of ‘reverse differentiation’ to include the

1
integration of eax + b , , sin(ax + b ) , cos(ax + b ) and
ax + b
sec 2 (ax + b )

Knowledge of the general method of integration by

substitution is not required

use trigonometrical relationships in carrying out integration

e.g. use of double-angle formulae to integrate sin2 x or

cos2 (2 x )

understand and use the trapezium rule to estimate the value

of a definite integral

Includes use of sketch graphs in simple cases to determine

whether the trapezium rule gives an over-estimate or an

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Topic You should be able to R A G Comments

under-estimate

2.6 Numerical locate approximately a root of an equation, by means of

solution of graphical considerations and/or searching for a sign change
equations
e.g. finding a pair of consecutive integers between which a
root lies

understand the idea of, and use the notation for, a sequence
of approximations which converges to a root of an equation

understand how a given simple iterative formula of the form

xn +1 = F( xn ) relates to the equation being solved, and use a
given iteration, or an iteration based on a given
rearrangement of an equation, to determine a root to a
prescribed degree of accuracy

Knowledge of the condition for convergence is not included,

but an understanding that an iteration may fail to converge is
expected

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Paper 3 Pure Mathematics 3

Topic You should be able to R A G Comments

3.1 Algebra understand the meaning of | x | , sketch the graph of y = |ax +

b| and use relations such as | a | = | b | ⇔ a 2 = b 2 and
| x − a | < b ⇔ a − b < x < a + b in the course of solving
equations and inequalities

Graphs of y = | f( x ) | and y = f(| x |) for non-linear functions f

are not included.

e.g. 3x − 2 = 2x + 7 , 2x + 5 < x + 1

divide a polynomial, of degree not exceeding 4, by a linear or

quadratic polynomial, and identify the quotient and remainder
(which may be zero)

use the factor theorem and the remainder theorem

e.g. to find factors and remainders, solve polynomial

equations or evaluate unknown coefficients. Factors of the
form (ax + b ) in which the coefficient of x is not unity are
included, and similarly for calculation of remainders

recall an appropriate form for expressing rational functions in

partial fractions, and carry out the decomposition, in cases
where the denominator is no more complicated than
• (ax + b )(cx + d )(ex + f ) ,
• (ax + b )(cx + d )2 ,
• (ax + b )(cx 2 + d )

Excluding cases where the degree of the numerator exceeds

that of the denominator

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Topic You should be able to R A G Comments

use the expansion of (1 + x )n , where n is a rational number

and | x | < 1

Finding the general term in an expansion is not included.

Adapting the standard series to expand e.g. ( 2 − 21 x )

−1
is
included, and determining the set of values of x for which
the expansion is valid in such cases is also included

3.2 Logarithmic understand the relationship between logarithms and indices,

and exponential and use the laws of logarithms (excluding change of base)
functions
understand the definition and properties of e x and ln x ,
including their relationship as inverse functions and their
graphs

Knowledge of the graph of y = ekx for both positive and

negative values of k is included

use logarithms to solve equations and inequalities in which

the unknown appears in indices

e.g. 2 x < 5 , 3 × 23 x −1 < 5 , 3 x +1 = 42 x −1

use logarithms to transform a given relationship to linear

form, and hence determine unknown constants by
considering the gradient and/or intercept

e.g. y = kx n , y = k (a x )

3.3 understand the relationship of the secant, cosecant and

Trigonometry cotangent functions to cosine, sine and tangent, and use

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Topic You should be able to R A G Comments

properties and graphs of all six trigonometric functions for

angles of any magnitude

use trigonometrical identities for the simplification and exact

evaluation of expressions and in the course of solving
equations, and select an identity or identities appropriate to
the context, showing familiarity in particular with the use of
• sec 2 θ ≡ 1 + tan2 θ and cosec 2 θ ≡ 1 + cot 2 θ ,
• the expansions of sin( A ± B ) , cos( A ± B ) and
tan( A ± B ) ,
• the formulae for sin 2A , cos 2A and tan 2A ,
• the expression of a sinθ + b cos θ in the forms
R sin(θ ± α ) and R cos(θ ± α )

e.g. simplifying cos( x − 30°) − 3 sin( x − 60°) .

e.g. solving tanθ + cot θ = 4 or 2 sec 2 θ − tanθ = 5 or

3 cos θ + 2 sinθ = 1

3.4 use the derivatives of e x , ln x , sin x , cos x , tan x and

Differentiation tan−1 x together with constant multiples, sums, differences
and composites

Derivatives of sin−1 x and cos−1 x not required

differentiate products and quotients

2x − 4 2
e.g. , x 2 ln x , x e1− x
3x + 2

find and use the first derivative of a function which is defined

parametrically or implicitly

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Topic You should be able to R A G Comments

e.g. x = t − e2t , y = t + e2t

e.g. x 2 + y 2 = xy + 7

Including use in problems involving tangents and normals

3.5 Integration extend the idea of ‘reverse differentiation’ to include the

1
integration of eax + b , , sin(ax + b ) , cos(ax + b ) and
ax + b
1
sec 2 (ax + b ) and 2
x + a2

1
Including examples such as
2 + 3x 2

use trigonometrical relationships in carrying out integration

e.g. use of double-angle formulae to integrate sin2 x or

cos2 (2 x )

integrate rational functions by means of decomposition into

partial fractions

Restricted to types of partial fractions as specified in section

1 above

k f ′( x )
recognise an integrand of the form , and integrate
f( x )
such functions

x
e.g. integration of or tan x
x2 + 1

recognise when an integrand can usefully be regarded as a

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Topic You should be able to R A G Comments

product, and use integration by parts

e.g. integration of x sin 2 x , x 2 e− x , ln x , x tan-1 x

use a given substitution to simplify and evaluate either a

definite or an indefinite integral

e.g. to integrate sin2 2 x cos x using the substitution u = sin x

3.6 Numerical locate approximately a root of an equation, by means of

solution of graphical considerations and/or searching for a sign change
equations
e.g. finding a pair of consecutive integers between which a
root lies

understand the idea of, and use the notation for, a sequence
of approximations which converges to a root of an equation

understand how a given simple iterative formula of the form

xn +1 = F( xn ) relates to the equation being solved, and use a
given iteration, or an iteration based on a given
rearrangement of an equation, to determine a root to a
prescribed degree of accuracy

Knowledge of the condition for convergence is not included,

but an understanding that an iteration may fail to converge is
expected

3.7 Vectors x
use standard notations for vectors, i.e. ()
x
y
, xi + yj , y ,
z
uur
xi + yj + zk , AB , a

carry out addition and subtraction of vectors and

multiplication of a vector by a scalar, and interpret these
operations in geometrical terms

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Topic You should be able to R A G Comments

uur uur uur

e.g. OABC is a parallelogram is equivalent to OB = OA + OC .
The general form of the ratio theorem is not included, but
understanding that the mid-point of AB has position vector
uur uur
1
2 (OA + OB ) is expected
calculate the magnitude of a vector, and use unit vectors,
displacement vectors and position vectors

In 2 or 3 dimensions

understand the significance of all the symbols used when the

equation of a straight line is expressed in the form r = a + tb ,
and find the equation of a line, given sufficient information

e.g. finding the equation of a line given the position vector of

a point on the line and a direction vector, or the position
vectors of two points on the line

determine whether two lines are parallel, intersect or are

skew, and find the point of intersection of two lines when it
exists

Calculation of the shortest distance between two skew lines

is excluded. Finding the equation of the common
perpendicular to two skew lines is also excluded

calculate the scalar product of two vectors, and use scalar

products in problems involving lines and points

e.g. finding the angle between two lines, & finding the foot of
the perpendicular from a point to a line; questions may
involve 3D objects such as cuboids, tetrahedra (pyramids),
etc.

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Knowledge of the vector product is not required

3.8 Differential formulate a simple statement involving a rate of change as a

equations differential equation

The introduction and evaluation of a constant of

proportionality, where necessary, is included

find by integration a general form of solution for a first order

differential equation in which the variables are separable

Any of the integration techniques from section 5 above may

be required

use an initial condition to find a particular solution

interpret the solution of a differential equation in the context

of a problem being modelled by the equation

Where a differential equation is used to model a ‘real-life’

situation, no specialised knowledge of the context will be
required

3.9 Complex understand the idea of a complex number, recall the meaning
numbers of the terms real part, imaginary part, modulus, argument,
conjugate, and use the fact that two complex numbers are
equal if and only if both real and imaginary parts are equal

Notations Re z , Im z , | z | , argz , z∗ should be known. (The

argument of a complex number will usually refer to an angle θ
such that −π < θ ≤ π , but in some cases the interval
0 ≤ θ < 2π may be more convenient. Answers may use
either interval unless the question specifies otherwise.)

carry out operations of addition, subtraction, multiplication

and division of two complex numbers expressed in Cartesian

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Topic You should be able to R A G Comments

form x + iy

For calculations involving multiplication or division, full details

of the working should be shown

use the result that, for a polynomial equation with real

coefficients, any non-real roots occur in conjugate pairs

e.g. in solving a cubic or quartic equation where one complex

root is given

represent complex numbers geometrically by means of an

Argand diagram

carry out operations of multiplication and division of two

complex numbers expressed in polar form
r (cos θ + isinθ ) ≡ r eiθ

The results | z1z2 | = | z1 || z2 | and

arg( z1z2 ) = arg( z1) + arg( z2 ) , and corresponding results for
division, are included

find the two square roots of a complex number

e.g. the square roots of 5 + 12i in exact Cartesian form. Full

details of the working should be shown

understand in simple terms the geometrical effects of

conjugating a complex number and of adding, subtracting,
multiplying and dividing two complex numbers

illustrate simple equations and inequalities involving complex

numbers by means of loci in an Argand diagram

e.g. | z − a | < k , | z − a | = | z − b | , arg( z − a ) = α

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Paper 4 Mechanics

Topic You should be able to R A G Comments

4.1 Forces and identify the forces acting in a given situation

equilibrium
e.g. by drawing a force diagramunderstand the vector nature
of force, and find and use components and resultants

Calculations are always required, not approximate solutions

by scale drawing

use the principle that, when a particle is in equilibrium, the

vector sum of the forces acting is zero, or equivalently, that
the sum of the components in any direction is zero

Solutions by resolving are usually expected, but equivalent

methods (e.g. triangle of forces, Lami’s Theorem, where
suitable) are entirely acceptable; any such additional
methods are not required knowledge, however, and will not
be referred to in questions

understand that a contact force between two surfaces can be

represented by two components, the normal component and
the frictional component

use the model of a ‘smooth’ contact, and understand the

limitations of this model

understand the concepts of limiting friction and limiting

equilibrium, recall the definition of coefficient of friction, and
use the relationship F = µ R or F ≤ µ R , as appropriate

Terminology such as ‘about to slip’ may be used to mean ‘in

limiting equilibrium’ in questions

use Newton’s third law

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Topic You should be able to R A G Comments

e.g. the force exerted by a particle on the ground is equal and

opposite to the force exerted by the ground on the particle

4.2 Kinematics understand the concepts of distance and speed as scalar

of motion in a quantities, and of displacement, velocity and acceleration as
straight line vector quantities

Restricted to motion in one dimension only. The term

‘deceleration’ may sometimes be used in the context of
decreasing speed

sketch and interpret displacement–time graphs and velocity–

time graphs, and in particular appreciate that
• the area under a velocity–time graph represents
displacement,
• the gradient of a displacement–time graph represents
velocity,
• the gradient of a velocity–time graph represents
acceleration

use differentiation and integration with respect to time to

solve simple problems concerning displacement, velocity and
acceleration

Calculus required is restricted to that within the content list for

component Pure Mathematics 1

use appropriate formulae for motion with constant

acceleration in a straight line

Problems may involve setting up more than one equation,

using information about the motion of different particles

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4.3 Momentum use the definition of linear momentum and show

understanding of its vector nature

For motion in one dimension only

use conservation of linear momentum to solve problems that

may be modelled as the direct impact of two bodies

The ‘direct impact’ of two bodies includes the case where the
bodies coalesce on impact. Knowledge of impulse and the
coefficient of restitution is not required

4.4 Newton’s apply Newton’s laws of motion to the linear motion of a

laws of motion particle of constant mass moving under the action of constant
forces, which may include friction, tension in an inextensible
string and thrust in a connecting rod

If any other forces resisting motion are to be considered (e.g.

air resistance) this will be indicated in the question

use the relationship between mass and weight

W = mg . In this component, questions are mainly numerical,

and use of the approximate numerical value 10 (m s–2) for g
is expected in such cases

solve simple problems which may be modelled as the motion

of a particle moving vertically or on an inclined plane with
constant acceleration

Including, for example, motion of a particle on a rough plane

where the acceleration while moving up the plane is different
from the acceleration while moving down the plane

solve simple problems which may be modelled as the motion

of connected particles

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e.g. particles connected by a light inextensible string passing

over a smooth pulley, or a car towing a trailer by means of
either a light rope or a light rigid tow-bar

4.5 Energy, understand the concept of the work done by a force, and
work and power calculate the work done by a constant force when its point of
application undergoes a displacement not necessarily parallel
to the force

W = Fd cos θ ; Use of the scalar product is not required

understand the concepts of gravitational potential energy and

kinetic energy, and use appropriate formulae

understand and use the relationship between the change in

energy of a system and the work done by the external forces,
and use in appropriate cases the principle of conservation of
energy

Including cases where the motion may not be linear (e.g. a

child on a smooth curved ‘slide’) so long as only overall
energy changes need to be considered

use the definition of power as the rate at which a force does

work, and use the relationship between power, force and
velocity for a force acting in the direction of motion

Work done
Including calculation of (average) power as .
Time taken

P = Fv .

solve problems involving, for example, the instantaneous

acceleration of a car moving on a hill against a resistance

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Paper 5 Probability & Statistics 1

Topic You should be able to R A G Comments

5.1 select a suitable way of presenting raw statistical data, and

Representation discuss advantages and/or disadvantages that particular
of data representations may have

construct and interpret stem-and-leaf diagrams, box-and-

whisker plots, histograms and cumulative frequency graphs

Including back-to-back stem-and-leaf diagrams

understand and use different measures of central tendency

(mean, median, mode) and variation (range, interquartile
range, standard deviation)

e.g. in comparing and contrasting sets of data

use a cumulative frequency graph

e.g. to estimate medians, quartiles, percentiles, the

proportion of a distribution above (or below) a given value, or
between two values

calculate and use the mean and standard deviation of a set

of data (including grouped data) either from the data itself or
from given totals Σx and Σx 2 , or coded totals Σ( x − a ) and
Σ( x − a )2 , and use such totals in solving problems which may
involve up to two data sets.

5.2 understand the terms permutation and combination, and

Permutations solve simple problems involving selections
and
combinations solve problems about arrangements of objects in a line,
including those involving

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• repetition (e.g. the number of ways of arranging the

letters of the word ‘NEEDLESS’),
• restriction (e.g. the number of ways several people can
stand in a line if two particular people must — or must
not — stand next to each other)

Problems may include cases such as people sitting in two (or

more) rows. Problems about objects arranged in a circle are
excluded

5.3 Probability evaluate probabilities in simple cases by means of

enumeration of equiprobable elementary events, or by
calculation using permutations or combinations

e.g. the total score when two fair dice are thrown.

e.g. drawing balls at random from a bag containing balls of

different colours

use addition and multiplication of probabilities, as

appropriate, in simple cases

Explicit use of the general formula

P( A ∪ B ) = P( A) + P(B ) − P( A ∩ B ) is not required

understand the meaning of exclusive and independent

events, including determination of whether events A and B
are independent by comparing the values of P( A ∩ B ) and
P( A) × P(B )

calculate and use conditional probabilities in simple cases

e.g. situations that can be represented by a sample space of

equiprobable elementary events, or a tree diagram. The use

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P( A ∩ B )
of P( A | B ) = may be required in simple cases
P(B )

construct a probability distribution table relating to a given

5.4 Discrete situation involving a discrete random variable X, and
random calculate E(X) and Var(X)
variables
use formulae for probabilities for the binomial and geometric
distributions, and recognise practical situations where these
distributions are suitable models

The notations B(n, p ) and Geo( p ) are included. Geo(p)

denotes the distribution in which pr = p(1 – p)r–1 for r = 1, 2, 3,

use formulae for the expectation and variance of the binomial

distribution and for the expectation of the geometric
distribution

Proofs of formulae are not required

5.5 The normal understand the use of a normal distribution to model a

distribution continuous random variable, and use normal distribution
tables

Sketches of normal curves to illustrate distributions or

probabilities may be required

solve problems concerning a variable X, where X ~ N( µ, σ 2 )

, including
• finding the value of P( X > x1 ) , or a related probability,
given the values of x1 , µ , σ ,
• finding a relationship between x1 , µ and σ given the
value of P( X > x1 ) or a related probability

recall conditions under which the normal distribution can be

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Topic You should be able to R A G Comments

used as an approximation to the binomial distribution, and

use this approximation, with a continuity correction, in solving
problems

n sufficiently large to ensure that both np > 5 and nq > 5

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Paper 6 Probability & Statistics 2

Topic You should be able to R A G Comments

6.1 The Poisson calculate probabilities for the distribution Po(λ )

distribution
use the fact that if X ~ Po(λ ) then the mean and variance of
X are each equal to λ

Proofs are not required

understand the relevance of the Poisson distribution to the

distribution of random events, and use the Poisson
distribution as a model

use the Poisson distribution as an approximation to the

binomial distribution where appropriate

The conditions that n is large and p is small should be known;

n > 50 and np < 5 , approximately

use the normal distribution, with continuity correction, as an

approximation to the Poisson distribution where appropriate

The condition that λ is large should be known; λ > 15 ,

approximately

6.2 Linear use, in the course of solving problems, the results that
combinations of • E(aX + b ) = a E( X ) + b and Var(aX + b ) = a 2 Var( X ) ,
random
• E(aX + bY ) = a E( X ) + b E(Y ) ,
variables
• Var(aX + bY ) = a 2 Var( X ) + b 2 Var(Y ) for
independent X and Y,
• if X has a normal distribution then so does aX + b ,
• if X and Y have independent normal distributions then
aX + bY has a normal distribution,

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• if X and Y have independent Poisson distributions then

X + Y has a Poisson distribution

Proofs of these results are not required

6.3 Continuous understand the concept of a continuous random variable, and

random recall and use properties of a probability density function
variables
For density functions defined over a single interval only; the
3
domain may be infinite, e.g. 4 for x ≥ 1
x

use a probability density function to solve problems involving

probabilities, and to calculate the mean and variance of a
distribution

Location of the median or other percentiles of a distribution

by direct consideration of an area using the density function
may be required but explicit knowledge of the cumulative
distribution function is not included

6.4 Sampling understand the distinction between a sample and a

and estimation population, and appreciate the necessity for randomness in
choosing samples

explain in simple terms why a given sampling method may be

unsatisfactory

Knowledge of particular sampling methods, such as quota or

stratified sampling, is not required, but an elementary
understanding of the use of random numbers in producing
random samples is included

recognise that a sample mean can be regarded as a random

variable, and use the facts that E( X ) = µ and that

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σ2
Var( X ) =
n

use the fact that X has a normal distribution if X has a

normal distribution

use the Central Limit Theorem where appropriate

Only an informal understanding of the Central Limit Theorem

(CLT) is required; for large sample sizes, the distribution of a
sample mean is approximately normal

calculate unbiased estimates of the population mean and

variance from a sample, using either raw or summarised data

Only a simple understanding of the term ‘unbiased’ is

required, e.g. that although individual estimates will vary the
process gives an accurate result ‘on average’

determine and interpret a confidence interval for a population

mean in cases where the population is normally distributed
with known variance or where a large sample is used

determine, from a large sample, an approximate confidence

interval for a population proportion

6.5 Hypothesis understand the nature of a hypothesis test, the difference

tests between one-tailed and two-tailed tests, and the terms null
hypothesis, alternative hypothesis, significance level,
rejection region (or critical region), acceptance region and
test statistic

Outcomes of hypothesis tests are expected to be interpreted

in terms of the contexts in which questions are set

formulate hypotheses and carry out a hypothesis test in the

context of a single observation from a population which has a

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binomial or Poisson distribution, using

• direct evaluation of probabilities,
• a normal approximation to the binomial or the Poisson
distribution, where appropriate

formulate hypotheses and carry out a hypothesis test

concerning the population mean in cases where the
population is normally distributed with known variance or
where a large sample is used

understand the terms Type I error and Type II error in relation

to hypothesis tests

calculate the probabilities of making Type I and Type II errors

in specific situations involving tests based on a normal
distribution or direct evaluation of binomial or Poisson
probabilities

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Section 8: Useful websites

The websites listed below are useful resources to help you study for your Cambridge International AS and A
Level Mathematics.

www.examsolutions.co.uk
Here you will find video tutorials on many of the topics in the Pure Mathematics, Probability & Statistics and
Mechanics papers. There are videos showing exam questions, some of which are from Cambridge
International papers but many others from UK syllabuses will be useful for revision.

www.bbc.co.uk/bitesize/higher/maths
This site has revision notes and examples on some of the Pure Mathematics 1 topics. (Note that this is not
accessible from all countries.)

www.s-cool.co.uk/a-level/maths
Good coverage of Pure Mathematics and Statistics topics, but not Mechanics. Revision material is arranged
by topic and includes explanations, revision summaries and exam-style questions with answers.

http://www.cimt.org.uk/
This site contains useful course materials – textbook style notes, worked examples and exercises – on Pure
Mathematics, Mechanics and Probability & Statistics topics. Look for the MEP link to access these. Answers
to the exercises are available to teachers who can request a password.

www.khanacademy.org
A site with many useful video explanations and some exercises. Uses American terminology but still helpful
for revision purposes. You can search by topic or by school grade (year).

www.mathcentre.ac.uk
Notes, examples and exercises for many of the topics on the syllabus.

www.waldomaths.com/index1619.jsp
This site provides Java applets (animations) on a variety of mathematics topics on the syllabus.

www.physicsandmathstutor.com
A source of useful revision notes, exercises and sets of questions from UK examination papers that have
been organised by difficulty.

https://nrich.maths.org/9088
Resources for learners aged 16+ from NRICH at Cambridge University.

https://mrbartonmaths.com/students/a-level
Notes, videos and examples on various AS and A Level topics. Also includes exam papers and mark
schemes for UK exam boards.

www.getrevising.co.uk/timetable
A study planner you can use to help you plan your revision.

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